Subgroups — #36379775

Order 22599.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419391	C₃	1	1	yes	maximal, characteristic
2	#21419232	C₃	1	2	yes	maximal, complemented
3	C₃²×C₂₇×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578912	—	3	1	no	—
5	#14578902	C₉	1	2	yes	—
6	#14578900	C₉	1	1	yes	characteristic
7	#14578897	—	3	1	no	—
8	#14578894	—	3	1	no	—
9	#14578892	—	3	2	no	—
10	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
11	C₃×C₂₇×C₃₁	—	3	1	no	—
12	C₃×C₂₇×C₃₁	—	3	3	no	—
13	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
14	C₃×C₉×C₃₁	—	3	1	no	—
15	C₃×C₉×C₃₁	—	3	1	no	—
16	C₃×C₉×C₃₁	—	3	1	no	—
17	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
21	#11719689	—	3	2	no	—
22	#11719687	—	9	1	no	—
23	#11719686	—	9	1	no	—
24	C₉×C₃₁⋊C₃	—	9	1	no	—
25	C₂₇×C₃₁	—	3	3	no	—
26	C₃₁⋊C₂₇	—	9	2	no	—
27	#10624526	—	31	1	no	maximal, Sylow
28	C₃²×C₃₁	—	3	3	no	—
29	C₃²×C₃₁	—	3	1	no	—
30	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
31	C₃×C₃₁⋊C₃	—	9	1	no	—
32	C₉×C₃₁	—	3	1	no	—
33	C₉×C₃₁	—	3	1	no	—
34	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₉×C₃₁	#731	1	1	yes	characteristic
36	C₉×C₃₁	#732	1	1	yes	characteristic
37	C₃₁⋊C₉	—	9	2	no	—
38	#6908	—	31	1	no	—
39	C₃²×C₂₇	C₃₁⋊C₃	1	1	yes	characteristic, complemented
40	#6869	—	31	2	no	—
41	C₃×C₃₁	—	3	3	no	—
42	C₃×C₃₁	—	3	1	no	—
43	C₃×C₃₁	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
44	C₃₁⋊C₃	—	27	1	no	—
45	#737	—	93	1	no	—
46	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
47	#733	—	31	2	no	—
48	#732	—	31	1	no	—
49	#731	—	93	1	no	—
50	AΣL(1,𝔽₂₇)	—	93	1	no	—
51	C₂₇⋊C₃	—	93	2	no	—
52	C₃×C₂₇	—	3	1	no	—
53	C₃×C₂₇	—	3	3	no	—
54	C₃₁	#10624526	1	1	yes	characteristic, complemented, Sylow
55	C₃³	#11719687	1	1	yes	characteristic
56	3₋¹⁺²	—	279	1	no	—
57	3₋¹⁺²	—	279	1	no	—
58	3₋¹⁺²	—	93	2	no	—
59	H(3,𝔽₃)	—	93	1	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	1	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	279	1	no	—
64	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
65	C₃×C₉	#11719686	1	1	yes	characteristic
66	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
67	C₂₇	—	3	3	no	—
68	C₂₇	—	279	2	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	279	1	no	—
72	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
73	C₉	—	3	1	no	—
74	C₉	—	3	1	no	—
75	C₉	—	279	2	no	—
76	C₉	#14578894	1	1	yes	characteristic
77	C₉	#14578897	1	1	yes	characteristic
78	C₉	#14578900	1	1	yes	characteristic
79	C₃	—	3	1	no	—
80	C₃	—	3	3	no	—
81	C₃	—	837	1	no	—
82	C₃	#21419235	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#14578900
commutator	C₃×C₉×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃²×C₂₇×C₃₁	C₃
socle	C₃×C₃₁	H(3,𝔽₃)⋊C₉
prime radical	#21419391	C₃
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	C₃²×C₂₇	C₃₁⋊C₃
hypocenter	C₃₁	#10624526
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²×C₂₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #21419232	C₃	semidirect
3. C₃²×C₂₇×C₃₁	C₃	semidirect
39. C₃²×C₂₇	C₃₁⋊C₃	semidirect
54. C₃₁	#10624526	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21419391	3
#21419232	3
C₃²×C₂₇×C₃₁	3
#10624526	31